In this final chapter, we give a more detailed account of the role played by the Axiom of Choice (AC) in the theory of paradoxical decompositions. Ever since its discovery, the Banach-Tarski Paradox has caused many mathematicians to look critically at the Axiom of Choice. Indeed, as soon as the Hausdorff Paradox was discovered it was challenged because of its use of AC; E. Borel [21, p. 256] objected because the choice set was not explicitly defined. We shall discuss these criticisms in more detail later in this chapter, but first we deal with several technical points that are essential to understanding the connection between AC and the Banach-Tarski Paradox.

Results of modern set theory can be used to show that AC is indeed necessary to obtain the Banach-Tarski Paradox, in the sense that the paradox is not a theorem of ZF alone. Before we can explain why this is so we need to introduce some notation and discuss some technical points of set theory. If T is a collection of sentences in the language of set theory, for example, T = ZF or T = ZF + AC, then Con(T) is the assertion, also a statement of set theory in fact, that T is consistent, that is, that a contradiction cannot be derived from T using the usual methods of proof. We take Con(ZF) as an underlying assumption in all that follows. Gödel proved in 1938 that Con(ZF) implies (and so is equivalent to) Con(ZF + AC); thus AC does not contradict ZF (see [98, 99]).